Expression of type Lambda

from the theory of proveit.linear_algebra.vector_sets

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, ExprRange, IndexedVar, K, Lambda, V, Variable, n, x
from proveit.core_expr_types import a_1_to_n, x_1_to_n
from proveit.linear_algebra import Span, lin_comb_axn
from proveit.logic import And, Equals, InSet, Set, SetOfAll
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda([x_1_to_n], Conditional(Equals(Span(Set(x_1_to_n)), SetOfAll(instance_param_or_params = [a_1_to_n], instance_element = lin_comb_axn, domain = K)).with_wrapping_at(2), And(ExprRange(sub_expr1, InSet(IndexedVar(x, sub_expr1), V), one, n))))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(x_{1}, x_{2}, \ldots, x_{n}\right) \mapsto \left\{\begin{array}{c} \begin{array}{l} \textrm{Span}\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\right) =  \\ \left\{\left(a_{1} \cdot x_{1}\right) +  \left(a_{2} \cdot x_{2}\right) +  \ldots +  \left(a_{n} \cdot x_{n}\right)\right\}_{a_{1}, a_{2}, \ldots, a_{n} \in K} \end{array} \end{array} \textrm{ if } \left(x_{1} \in V\right) \land  \left(x_{2} \in V\right) \land  \ldots \land  \left(x_{n} \in V\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 19 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 32 operands: 6
4	Literal
5	ExprTuple	7, 8
6	ExprTuple	9
7	Operation	operator: 10 operand: 15
8	Operation	operator: 12 operand: 16
9	ExprRange	lambda_map: 14 start_index: 38 end_index: 39
10	Literal
11	ExprTuple	15
12	Literal
13	ExprTuple	16
14	Lambda	parameter: 52 body: 17
15	Operation	operator: 18 operands: 19
16	Lambda	parameters: 20 body: 21
17	Operation	operator: 44 operands: 22
18	Literal
19	ExprTuple	23
20	ExprTuple	24
21	Conditional	value: 25 condition: 26
22	ExprTuple	46, 27
23	ExprRange	lambda_map: 28 start_index: 38 end_index: 39
24	ExprRange	lambda_map: 29 start_index: 38 end_index: 39
25	Operation	operator: 30 operands: 31
26	Operation	operator: 32 operands: 33
27	Variable
28	Lambda	parameter: 52 body: 46
29	Lambda	parameter: 52 body: 47
30	Literal
31	ExprTuple	34
32	Literal
33	ExprTuple	35
34	ExprRange	lambda_map: 36 start_index: 38 end_index: 39
35	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
36	Lambda	parameter: 52 body: 40
37	Lambda	parameter: 52 body: 41
38	Literal
39	Variable
40	Operation	operator: 42 operands: 43
41	Operation	operator: 44 operands: 45
42	Literal
43	ExprTuple	47, 46
44	Literal
45	ExprTuple	47, 48
46	IndexedVar	variable: 49 index: 52
47	IndexedVar	variable: 50 index: 52
48	Variable
49	Variable
50	Variable
51	ExprTuple	52
52	Variable